In this paper we study the compressible finite extensible nonlinear elastic (FENE) dumbbell model [1]}, [2]}:
\(\left\lbrace \begin{array}{ll}\varrho _t+div(\varrho u)=0 , \\[1ex](\varrho u)_t+div(\varrho u\otimes u)-div\Sigma {(u)}+\nabla _x{P(\varrho )}=div~\tau , \\[1ex]\psi _t+u\cdot \nabla \psi =div_{R}[- \sigma (u)\cdot {R}\psi +\nabla _{R}\psi +\nabla _{R}\mathcal {U}\psi ], \\[1ex]\tau _{ij}=\int _{B}(R_{i}\nabla _{j}\mathcal {U})\psi dR, \\[1ex]\varrho |_{t=0}=\varrho _0,~~u|_{t=0}=u_0,~~\psi |_{t=0}=\psi _0, \\[1ex](\nabla _{R}\psi +\nabla _{R}\mathcal {U}\psi )\cdot {n}=0 ~~~~ \text{on} ~~~~ \partial B(0,R_{0}) .\\[1ex]\end{array}\right.\) 
